Shen Qu, 918881147

Discussion

• Indepency and grouping

Linear regression is one of the Watershed Hydrological Model. Singh (1972) used linear models with a logarithm transformation of the variables.

\[\ln(Dependent\ Variable)=β_0+β_1\ln P+β_2\ln Q+β_3\ln F_P+β_4CC+Interactions+\varepsilon\]

where the dependent variables can either be total storm flow volume (\(Q_t\)) in mm, quick flow volume (\(Q_f\)) in mm or peak flow (\(Q_{pk}\)) in \(m^3 sec^{−1} km^{−2}\). Independent variables were storm rainfall (\(P\)) in \(mm\), initial flow (\(Q_i\)) in \(mm h^{−1}\), rainfall frequency (\(F_p\)), the inverse of rainfall duration, in \(h^{−1}\) and a dummy variable (\(CC\)) representing the treatment effect on basin. \(CC\) was 0 and 1 for the calibration (1967–1992) and treated (1994–1998) periods, respectively. \(β_0\) to \(β_4\) are regression coefficients of the independent variables. All interactions between the independent variables were also tested for significance at \(α=0.10\). (Guillemette et al., 2005)

Inspired by this theory, I divided the variables to 4 groups:

It is resonable that the best model should contains at least one variable in each group. In this way, X8 and X9 are indispensable variables. We have also know that Setpwise forward method could neglect this context and is not recommended for this question. When we try to simplify a model, we should keep each catergory being not empty and evaluate one variable with others inside the group.

• Log and interaction

It is obviously that the model of all-log with interaction is overfitting. Therefore, we should control the number of predictors. Since there are 6 predictors by elimination regression. A new model with higher R-squared and less than 7 predictors is better than the solution in question (7). After comparing other coefficients and plots, the The best model should be 1-3-6-8-9 all-log model.

\[\hat{\ln(y)}=5.87592+0.68118\ln(X_1)+0.37527\ln(X_3)-0.33576\ln(X_6)+1.72493\ln(X_8)-1.45653\ln(X_9)\]

If we removed X1 in the question (3), we cannot get this one.

I tried other combination mixed log and interaction, the results is interesting. For example, single variable model (log(X4)) has higher R-squared than 3-4-7 model. A mixed combination of X1:X3,X1:X4,ln(X8),ln(X9) with 5 variables has higher R-squared than 3-4-6-7-8-9 model. Due to time constraints, I didn’t tied all the possible combinations. It also might has a better answer if we use glm.

• Reference

[1]: Montgomery, D. C., Peck, E. A., & Vining, G. G. (2012). Introduction to linear regression analysis (Vol. 821). John Wiley & Sons.

[2]: Guillemette, F., Plamondon, A. P., Prévost, M., & Lévesque, D. (2005). Rainfall generated stormflow response to clearcutting a boreal forest: peak flow comparison with 50 world-wide basin studies. Journal of hydrology, 302(1-4), 137-153.

(a) The matrix of scatterplots and the correlation matrix

library(tidyverse)
library(GGally)
library(olsrr)
library(car)
library(MASS)
library(huxtable)
table_wf <- read_table2("WaterFlow.txt")

ggpairs(data=table_wf[c(1:10)])

(c) The fitted full model

# build the model
model_wf_full <- lm(y ~ X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9, data=table_wf)
model_wf_full%>% summary()

## 
## Call:
## lm(formula = y ~ X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9, 
##     data = table_wf)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -1404.21  -318.77    74.73   266.66  1274.30 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)  
## (Intercept)   292.56    4428.62   0.066   0.9480  
## X1           -203.14     410.27  -0.495   0.6259  
## X2           1055.78    9833.70   0.107   0.9156  
## X3            -49.24     156.20  -0.315   0.7558  
## X4            209.76     162.05   1.294   0.2103  
## X5            -10.20      51.09  -0.200   0.8438  
## X6            -24.56     303.53  -0.081   0.9363  
## X7            142.78    3288.44   0.043   0.9658  
## X8            511.71     209.74   2.440   0.0241 *
## X9           -301.87     172.00  -1.755   0.0945 .
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 609.3 on 20 degrees of freedom
## Multiple R-squared:  0.8214, Adjusted R-squared:  0.741 
## F-statistic: 10.22 on 9 and 20 DF,  p-value: 9.744e-06

Anova(model_wf_full)

(c) ii Residual diagnostics

#Model Fit Assessment
ols_plot_diagnostics(model_wf_full)

# Part & Partial Correlations
ols_test_correlation(model_wf_full) # Correlation between observed residuals and expected residuals under normality.

## [1] 0.9710713

# Residual Normality Test
ols_test_normality(model_wf_full) # Test for detecting violation of normality assumption. #If p-value is bigger, then no problem of non-normality #

## -----------------------------------------------
##        Test             Statistic       pvalue  
## -----------------------------------------------
## Shapiro-Wilk              0.9589         0.2898 
## Kolmogorov-Smirnov        0.1423         0.5314 
## Cramer-von Mises          2.5333         0.0000 
## Anderson-Darling          0.5169         0.1748 
## -----------------------------------------------

(c) iii The partial regression and nonlinear diagnostics

#Lack of Fit F Test
# ols_pure_error_anova(lm(y~X8, data = table_wf))

# Variable Contributions
ols_plot_added_variable(model_wf_full)

# Residual Plus Component Plot
ols_plot_comp_plus_resid(model_wf_full)

(c) iv Collinearity diagnostics

# for full model
ols_vif_tol(model_wf_full)

(d) The fitted log y model

# build full log model

 table_wf_logy <- table_wf %>% mutate(logy=log(y))
 table_wf_logy$y <- NULL
 ggpairs(data=table_wf_logy[c(1:10)])

model_wf_full_log <- lm(log(y) ~ X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9, data=table_wf)
summary(model_wf_full_log)

## 
## Call:
## lm(formula = log(y) ~ X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + 
##     X9, data = table_wf)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -0.95298 -0.20764  0.01499  0.18100  0.67539 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  3.402256   3.150312   1.080 0.293006    
## X1          -0.013532   0.291845  -0.046 0.963477    
## X2          -1.023664   6.995235  -0.146 0.885120    
## X3           0.177966   0.111113   1.602 0.124908    
## X4           0.108788   0.115272   0.944 0.356560    
## X5          -0.009622   0.036341  -0.265 0.793898    
## X6          -0.389474   0.215916  -1.804 0.086345 .  
## X7           4.233475   2.339245   1.810 0.085387 .  
## X8           0.630070   0.149200   4.223 0.000418 ***
## X9          -0.462276   0.122350  -3.778 0.001181 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.4334 on 20 degrees of freedom
## Multiple R-squared:  0.9474, Adjusted R-squared:  0.9237 
## F-statistic:    40 on 9 and 20 DF,  p-value: 7.513e-11

Anova(model_wf_full)

#Model Fit Assessment
ols_plot_diagnostics(model_wf_full_log)

# Part & Partial Correlations
ols_test_correlation(model_wf_full_log) # Correlation between observed residuals and expected residuals under normality.

## [1] 0.9808603

# Residual Normality Test
ols_test_normality(model_wf_full_log) # Test for detecting violation of normality assumption. #If p-value is bigger, then no problem of non-normality #

## -----------------------------------------------
##        Test             Statistic       pvalue  
## -----------------------------------------------
## Shapiro-Wilk              0.9659         0.4341 
## Kolmogorov-Smirnov        0.0993         0.9007 
## Cramer-von Mises          4.948          0.0000 
## Anderson-Darling          0.3686         0.4062 
## -----------------------------------------------

(d) (2) Collinearity diagnostics

# remove 1-9
model_wf_rm1_log <- lm(log(y) ~ X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9, data=table_wf)
model_wf_rm2_log <- lm(log(y) ~ X1 + X3 + X4 + X5 + X6 + X7 + X8 + X9, data=table_wf)
model_wf_rm3_log <- lm(log(y) ~ X1 + X2 + X4 + X5 + X6 + X7 + X8 + X9, data=table_wf)
model_wf_rm4_log <- lm(log(y) ~ X1 + X2 + X3 + X5 + X6 + X7 + X8 + X9, data=table_wf)
model_wf_rm5_log <- lm(log(y) ~ X1 + X2 + X3 + X4 + X6 + X7 + X8 + X9, data=table_wf)
model_wf_rm6_log <- lm(log(y) ~ X1 + X2 + X3 + X4 + X5 + X7 + X8 + X9, data=table_wf)
model_wf_rm7_log <- lm(log(y) ~ X1 + X2 + X3 + X4 + X5 + X6 + X8 + X9, data=table_wf)
model_wf_rm8_log <- lm(log(y) ~ X1 + X2 + X3 + X4 + X5 + X6 + X7 + X9, data=table_wf)
model_wf_rm9_log <- lm(log(y) ~ X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8, data=table_wf)

# Compare vif
ols_vif_tol(model_wf_full_log)

ols_vif_tol(model_wf_aic_log)

ols_vif_tol(model_wf_rm1_log)

ols_vif_tol(model_wf_rm1_aic_log)

ols_vif_tol(model_wf_rm2_log)

ols_vif_tol(model_wf_rm2_aic_log)

ols_vif_tol(model_wf_rm3_log)

ols_vif_tol(model_wf_rm3_aic_log)

ols_vif_tol(model_wf_rm4_log)

ols_vif_tol(model_wf_rm4_aic_log)

ols_vif_tol(model_wf_rm5_log)

ols_vif_tol(model_wf_rm5_aic_log)

ols_vif_tol(model_wf_rm6_log)

ols_vif_tol(model_wf_rm6_aic_log)

ols_vif_tol(model_wf_rm7_log)

ols_vif_tol(model_wf_rm7_aic_log)

ols_vif_tol(model_wf_rm8_log)

ols_vif_tol(model_wf_rm8_aic_log)

ols_vif_tol(model_wf_rm9_log)

ols_vif_tol(model_wf_rm9_aic_log)

(d) (3) Variable selection

# Removing one variable
huxreg(model_wf_rm1_log, model_wf_rm2_log, model_wf_rm3_log, model_wf_rm4_log, model_wf_rm5_log, model_wf_rm6_log, model_wf_rm7_log, model_wf_rm8_log, model_wf_rm9_log, model_wf_full_log)

# The results of elimination regression
huxreg(model_wf_rm1_aic_log, model_wf_rm2_aic_log, model_wf_rm3_aic_log, model_wf_rm4_aic_log, model_wf_rm5_aic_log, model_wf_rm6_aic_log, model_wf_rm7_aic_log, model_wf_rm8_aic_log, model_wf_rm9_aic_log, model_wf_aic_log)

(d) (4) Forward selection

Stepwise Forward Regression for full model

# Stepwise Forward Regression based on p values (use a=0.15) #
ols_step_forward_p(model_wf_full_log, penter = 0.15)

## Forward Selection Method    
## ---------------------------
## 
## Candidate Terms: 
## 
## 1. X1 
## 2. X2 
## 3. X3 
## 4. X4 
## 5. X5 
## 6. X6 
## 7. X7 
## 8. X8 
## 9. X9 
## 
## We are selecting variables based on p value...
## 
## Variables Entered: 
## 
## - X4 
## - X3 
## - X7 
## 
## No more variables to be added.
## 
## Final Model Output 
## ------------------
## 
##                         Model Summary                         
## -------------------------------------------------------------
## R                       0.944       RMSE               0.549 
## R-Squared               0.890       Coef. Var          8.618 
## Adj. R-Squared          0.878       MSE                0.301 
## Pred R-Squared          0.854       MAE                0.414 
## -------------------------------------------------------------
##  RMSE: Root Mean Square Error 
##  MSE: Mean Square Error 
##  MAE: Mean Absolute Error 
## 
##                                ANOVA                                
## -------------------------------------------------------------------
##                Sum of                                              
##               Squares        DF    Mean Square      F         Sig. 
## -------------------------------------------------------------------
## Regression     63.565         3         21.188    70.378    0.0000 
## Residual        7.828        26          0.301                     
## Total          71.393        29                                    
## -------------------------------------------------------------------
## 
##                                  Parameter Estimates                                  
## -------------------------------------------------------------------------------------
##       model     Beta    Std. Error    Std. Beta      t       Sig      lower    upper 
## -------------------------------------------------------------------------------------
## (Intercept)    2.872         0.547                 5.254    0.000     1.748    3.995 
##          X4    0.122         0.033        0.559    3.730    0.001     0.055    0.189 
##          X3    0.168         0.040        0.435    4.165    0.000     0.085    0.251 
##          X7    3.106         1.537        0.309    2.021    0.054    -0.053    6.266 
## -------------------------------------------------------------------------------------

## 
##                            Selection Summary                             
## ------------------------------------------------------------------------
##         Variable                  Adj.                                      
## Step    Entered     R-Square    R-Square     C(p)        AIC       RMSE     
## ------------------------------------------------------------------------
##    1    X4            0.8030      0.7960    48.8552    68.4060    0.7087    
##    2    X3            0.8731      0.8637    24.2129    57.2082    0.5792    
##    3    X7            0.8904      0.8777    19.6668    54.8305    0.5487    
## ------------------------------------------------------------------------

# Stepwise AIC Forward Regression #
ols_step_forward_aic(model_wf_full_log)

## Forward Selection Method 
## ------------------------
## 
## Candidate Terms: 
## 
## 1 . X1 
## 2 . X2 
## 3 . X3 
## 4 . X4 
## 5 . X5 
## 6 . X6 
## 7 . X7 
## 8 . X8 
## 9 . X9 
## 
## 
## Variables Entered: 
## 
## - X4 
## - X3 
## - X7 
## - X8 
## - X9 
## - X6 
## 
## No more variables to be added.

## 
##                        Selection Summary                        
## ---------------------------------------------------------------
## Variable      AIC      Sum Sq     RSS       R-Sq      Adj. R-Sq 
## ---------------------------------------------------------------
## X4           68.406    57.330    14.063    0.80302      0.79599 
## X3           57.208    62.335     9.057    0.87313      0.86373 
## X7           54.830    63.565     7.828    0.89036      0.87771 
## X8           54.522    64.144     7.248    0.89848      0.88223 
## X9           44.504    66.537     4.856    0.93199      0.91782 
## X6           39.161    67.591     3.801    0.94675      0.93286 
## ---------------------------------------------------------------

Stepwise Forward Regression for X4 eliminated model

# Stepwise Forward Regression based on p values (use a=0.15) #
ols_step_forward_p(model_wf_rm4_log, penter = 0.15)
# Stepwise AIC Forward Regression #
ols_step_forward_aic(model_wf_rm4_log)

Stepwise Forward Regression for X1 eliminated model

# Stepwise Forward Regression based on p values (use a=0.15) #
ols_step_forward_p(model_wf_rm1_log, penter = 0.15)
# Stepwise AIC Forward Regression #
ols_step_forward_aic(model_wf_rm1_log)

(d) (5) Backward selection

Stepwise Backward Regression for full model

# Stepwise Backward Regression based on p values (use a=0.05) #
ols_step_backward_p(model_wf_full_log, penter = 0.05)

## Backward Elimination Method 
## ---------------------------
## 
## Candidate Terms: 
## 
## 1 . X1 
## 2 . X2 
## 3 . X3 
## 4 . X4 
## 5 . X5 
## 6 . X6 
## 7 . X7 
## 8 . X8 
## 9 . X9 
## 
## We are eliminating variables based on p value...
## 
## Variables Removed: 
## 
## - X1 
## - X2 
## - X5 
## 
## No more variables satisfy the condition of p value = 0.3
## 
## 
## Final Model Output 
## ------------------
## 
##                         Model Summary                         
## -------------------------------------------------------------
## R                       0.973       RMSE               0.407 
## R-Squared               0.947       Coef. Var          6.385 
## Adj. R-Squared          0.933       MSE                0.165 
## Pred R-Squared          0.908       MAE                0.273 
## -------------------------------------------------------------
##  RMSE: Root Mean Square Error 
##  MSE: Mean Square Error 
##  MAE: Mean Absolute Error 
## 
##                                ANOVA                                
## -------------------------------------------------------------------
##                Sum of                                              
##               Squares        DF    Mean Square      F         Sig. 
## -------------------------------------------------------------------
## Regression     67.591         6         11.265     68.16    0.0000 
## Residual        3.801        23          0.165                     
## Total          71.393        29                                    
## -------------------------------------------------------------------
## 
##                                   Parameter Estimates                                    
## ----------------------------------------------------------------------------------------
##       model      Beta    Std. Error    Std. Beta      t        Sig      lower     upper 
## ----------------------------------------------------------------------------------------
## (Intercept)     2.692         0.445                  6.046    0.000     1.771     3.613 
##          X3     0.184         0.032        0.476     5.698    0.000     0.117     0.251 
##          X4     0.109         0.026        0.499     4.244    0.000     0.056     0.162 
##          X6    -0.368         0.146       -0.133    -2.526    0.019    -0.669    -0.066 
##          X7     4.085         1.213        0.406     3.367    0.003     1.575     6.595 
##          X8     0.612         0.133        0.493     4.614    0.000     0.337     0.886 
##          X9    -0.448         0.108       -0.450    -4.135    0.000    -0.672    -0.224 
## ----------------------------------------------------------------------------------------

## 
## 
##                           Elimination Summary                           
## -----------------------------------------------------------------------
##         Variable                  Adj.                                     
## Step    Removed     R-Square    R-Square     C(p)       AIC       RMSE     
## -----------------------------------------------------------------------
##    1    X1            0.9474      0.9273    8.0021    42.8146    0.4230    
##    2    X2            0.9472      0.9304    6.0604    40.9019    0.4139    
##    3    X5            0.9468      0.9329    4.2345    39.1611    0.4065    
## -----------------------------------------------------------------------

# Stepwise AIC Backward Regression #
ols_step_backward_aic(model_wf_full_log)

## Backward Elimination Method 
## ---------------------------
## 
## Candidate Terms: 
## 
## 1 . X1 
## 2 . X2 
## 3 . X3 
## 4 . X4 
## 5 . X5 
## 6 . X6 
## 7 . X7 
## 8 . X8 
## 9 . X9 
## 
## 
## Variables Removed: 
## 
## - X1 
## - X2 
## - X5 
## 
## No more variables to be removed.

## 
## 
##                   Backward Elimination Summary                   
## ---------------------------------------------------------------
## Variable       AIC       RSS     Sum Sq     R-Sq      Adj. R-Sq 
## ---------------------------------------------------------------
## Full Model    44.811    3.757    67.635    0.94737      0.92369 
## X1            42.815    3.758    67.635    0.94737      0.92731 
## X2            40.902    3.769    67.624    0.94721      0.93042 
## X5            39.161    3.801    67.591    0.94675      0.93286 
## ---------------------------------------------------------------

Stepwise Backward Regression for X4 eliminated model

# Stepwise Backward Regression based on p values (use a=0.05) #
ols_step_backward_p(model_wf_rm4_log, penter = 0.05)
# Stepwise AIC Backward Regression #
ols_step_backward_aic(model_wf_rm4_log)

Stepwise Backward Regression for X1 eliminated model

# Stepwise Backward Regression based on p values (use a=0.05) #
ols_step_backward_p(model_wf_rm1_log, penter = 0.05)
# Stepwise AIC Backward Regression #
ols_step_backward_aic(model_wf_rm1_log)

(d) (6) Best Subset Regression

# For full model #
k <- ols_step_best_subset(model_wf_full_log)
k

plot(k)

# For X4 eliminated model #
# k <- ols_step_best_subset(model_wf_rm4_log)
# k
# plot(k)

# For X1 eliminated model #
# k <- ols_step_best_subset(model_wf_rm1_log)
# k
# plot(k)

(d) (7) Models Comparison

• Model 437896

# build model 437896
model_wf_437896_log <- lm(log(y) ~ X4 + X3 + X7 + X8 + X9 + X6, data=table_wf)
ols_regress(model_wf_437896_log)

##                         Model Summary                         
## -------------------------------------------------------------
## R                       0.973       RMSE               0.407 
## R-Squared               0.947       Coef. Var          6.385 
## Adj. R-Squared          0.933       MSE                0.165 
## Pred R-Squared          0.908       MAE                0.273 
## -------------------------------------------------------------
##  RMSE: Root Mean Square Error 
##  MSE: Mean Square Error 
##  MAE: Mean Absolute Error 
## 
##                                ANOVA                                
## -------------------------------------------------------------------
##                Sum of                                              
##               Squares        DF    Mean Square      F         Sig. 
## -------------------------------------------------------------------
## Regression     67.591         6         11.265     68.16    0.0000 
## Residual        3.801        23          0.165                     
## Total          71.393        29                                    
## -------------------------------------------------------------------
## 
##                                   Parameter Estimates                                    
## ----------------------------------------------------------------------------------------
##       model      Beta    Std. Error    Std. Beta      t        Sig      lower     upper 
## ----------------------------------------------------------------------------------------
## (Intercept)     2.692         0.445                  6.046    0.000     1.771     3.613 
##          X4     0.109         0.026        0.499     4.244    0.000     0.056     0.162 
##          X3     0.184         0.032        0.476     5.698    0.000     0.117     0.251 
##          X7     4.085         1.213        0.406     3.367    0.003     1.575     6.595 
##          X8     0.612         0.133        0.493     4.614    0.000     0.337     0.886 
##          X9    -0.448         0.108       -0.450    -4.135    0.000    -0.672    -0.224 
##          X6    -0.368         0.146       -0.133    -2.526    0.019    -0.669    -0.066 
## ----------------------------------------------------------------------------------------

Anova(model_wf_437896_log)

confint(model_wf_437896_log, level = 1-(0.05/7)) # Bonferroni joint confidence interval #

##                 0.357 %    99.643 %
## (Intercept)  1.37732232  4.00627202
## X4           0.03318876  0.18491189
## X3           0.08857700  0.27911109
## X7           0.50312634  7.66681371
## X8           0.22022202  1.00298907
## X9          -0.76727751 -0.12799849
## X6          -0.79716475  0.06213151

# Collinearity Diagnostics #
ols_vif_tol(model_wf_437896_log)

#Model Fit Assessment
ols_plot_diagnostics(model_wf_437896_log)

# Part & Partial Correlations
ols_test_correlation(model_wf_437896_log) # Correlation between observed residuals and expected residuals under normality.

## [1] 0.9837263

# Residual Normality Test
ols_test_normality(model_wf_437896_log) # Test for detecting violation of normality assumption. #If p-value is bigger, then no problem of non-normality #

## -----------------------------------------------
##        Test             Statistic       pvalue  
## -----------------------------------------------
## Shapiro-Wilk              0.9728         0.6175 
## Kolmogorov-Smirnov        0.0997         0.8982 
## Cramer-von Mises          4.8429         0.0000 
## Anderson-Darling          0.2996         0.5612 
## -----------------------------------------------

# Variable Contributions
ols_plot_added_variable(model_wf_437896_log)

# Residual Plus Component Plot
ols_plot_comp_plus_resid(model_wf_437896_log)

• Model 437

# build model 437
model_wf_437_log <- lm(log(y) ~ X4 + X3 + X7, data=table_wf)
ols_regress(model_wf_437_log)

##                         Model Summary                         
## -------------------------------------------------------------
## R                       0.944       RMSE               0.549 
## R-Squared               0.890       Coef. Var          8.618 
## Adj. R-Squared          0.878       MSE                0.301 
## Pred R-Squared          0.854       MAE                0.414 
## -------------------------------------------------------------
##  RMSE: Root Mean Square Error 
##  MSE: Mean Square Error 
##  MAE: Mean Absolute Error 
## 
##                                ANOVA                                
## -------------------------------------------------------------------
##                Sum of                                              
##               Squares        DF    Mean Square      F         Sig. 
## -------------------------------------------------------------------
## Regression     63.565         3         21.188    70.378    0.0000 
## Residual        7.828        26          0.301                     
## Total          71.393        29                                    
## -------------------------------------------------------------------
## 
##                                  Parameter Estimates                                  
## -------------------------------------------------------------------------------------
##       model     Beta    Std. Error    Std. Beta      t       Sig      lower    upper 
## -------------------------------------------------------------------------------------
## (Intercept)    2.872         0.547                 5.254    0.000     1.748    3.995 
##          X4    0.122         0.033        0.559    3.730    0.001     0.055    0.189 
##          X3    0.168         0.040        0.435    4.165    0.000     0.085    0.251 
##          X7    3.106         1.537        0.309    2.021    0.054    -0.053    6.266 
## -------------------------------------------------------------------------------------

# Collinearity Diagnostics #
ols_vif_tol(model_wf_437_log)

#Model Fit Assessment
ols_plot_diagnostics(model_wf_437_log)

# Part & Partial Correlations
ols_test_correlation(model_wf_437_log) # Correlation between observed residuals and expected residuals under normality.

## [1] 0.9856766

# Residual Normality Test
ols_test_normality(model_wf_437_log) # Test for detecting violation of normality assumption. #If p-value is bigger, then no problem of non-normality #

## -----------------------------------------------
##        Test             Statistic       pvalue  
## -----------------------------------------------
## Shapiro-Wilk              0.9765         0.7267 
## Kolmogorov-Smirnov        0.1033         0.8736 
## Cramer-von Mises          3.1908         0.0000 
## Anderson-Darling          0.3511         0.4469 
## -----------------------------------------------

# Variable Contributions
ols_plot_added_variable(model_wf_437_log)

# Residual Plus Component Plot
ols_plot_comp_plus_resid(model_wf_437_log)

# Check PRESS Statistic
ols_press(model_wf_full)

## [1] 15880486

ols_press(model_wf_full_log)

## [1] 8.136733

ols_press(model_wf_437896_log)

## [1] 6.538275

ols_press(model_wf_437_log)

## [1] 10.43262

# ols_press(model_wf_137689_log)

# prediction power
ols_pred_rsq(model_wf_full)

## [1] 0.6179649

ols_pred_rsq(model_wf_full_log)

## [1] 0.8860283

ols_pred_rsq(model_wf_437896_log)

## [1] 0.908418

ols_pred_rsq(model_wf_437_log)

## [1] 0.8538697

# ols_pred_rsq(model_wf_137689_log)

More

• Other Models

# build X1*X8 eliminated log model
model_wf_18rm4_log <- lm(log(y) ~ X1*X8 + X3 + X6 + X7 + X9, data=table_wf)

# build X1*X8 eliminated log model
table_wf_resi <- table_wf%>% mutate(x1t8=X1*X8)
model_wf_1time8_log <- lm(log(y) ~ x1t8 + X3 + X6 + X7+ X9 , data=table_wf_resi)

# build X1*X4 eliminated log model
table_wf_resi <- table_wf%>% mutate(x1t4=X1*X4)
model_wf_1time4_log <- lm(log(y) ~ x1t4 + X3 + X6 + X7+ X8+ X9, data=table_wf_resi)
summary(model_wf_1time4_log)

# build X1/X4 eliminated log model
table_wf_resi <- table_wf%>% mutate(x14=X1/X4)
model_wf_1per4_log <- lm(log(y) ~ x14 + X3 + X6 + X7+ X8+ X9, data=table_wf_resi)

# build X4*X3 eliminated log model
model_wf_43rm1_log <- lm(log(y) ~ X9 + X4*X3 + X6 + X7 + X8 , data=table_wf)

# build X4*X9 eliminated log model
model_wf_49rm1_log <- lm(log(y) ~ X3 + X4*X9 + X6 + X7 + X8 , data=table_wf)

# build X4*X9 eliminated log model
model_wf_48rm1_log <- lm(log(y) ~ X3 + X4*X8 + X6 + X7 + X9 , data=table_wf)

# build X4*X9 eliminated log model
model_wf_47rm1_log <- lm(log(y) ~ X3 + X4*X7 + X6 + X9 + X8 , data=table_wf)

# build X4/X9 eliminated log model
table_wf_resi <- table_wf%>% mutate(x4p9=X4/X9)
model_wf_4per9_log <- lm(log(y) ~ X3 + x4p9 + X6 + X7 + X8 , data=table_wf_resi)

# build X3/X4vX8*X9 eliminated log model
model_wf_34v89_log <- lm(log(y) ~ X3*X4 + X8*X9 + X6 + X7, data=table_wf_resi)

# build X3/X4vX8*X9 eliminated log model
model_wf_34v89v67_log <- lm(log(y) ~ X3*X4 + X8*X9 + X6*X7, data=table_wf_resi)

# build X8/X9vX4*X3 eliminated log model
table_wf_resi <- table_wf%>% mutate(x8p9=X8/X9)
model_wf_8per9v43_log <- lm(log(y) ~ x8p9 + X4*X3 + X6 + X7, data=table_wf_resi)

# build X6/7vX8/X9vX4X3 eliminated log model
table_wf_resi <- table_wf%>% mutate(x8p9=X8/X9,x6p7=X6/X7)
model_wf_6p7v8p9v43_log <- lm(log(y) ~ x8p9 + X4*X3 + x6p7, data=table_wf_resi)

# build X8/X9vX4*X3rmX7 eliminated log model
table_wf_resi <- table_wf%>% mutate(x8p9=X8/X9)
model_wf_8per9v43rm7_log <- lm(log(y) ~ x8p9 + X4*X3 + X6, data=table_wf_resi)

# build X8/X9vX4*X3vX6/X7 eliminated log model
table_wf_resi <- table_wf%>% mutate(x8p9=X8/X9)
model_wf_8per9v43rm7_log <- lm(log(y) ~ x8p9 + X4*X3 + X6, data=table_wf_resi)

huxreg(model_wf_8per9v43rm7_log, model_wf_8per9v43_log, model_wf_43rm1_log, model_wf_6p7v8p9v43_log, model_wf_34v89_log, model_wf_34v89v67_log)

• Interaction models

# Interaction regression for full interaction
model_wf_full_log_inter <- lm(log(y)~ (X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9)^2, data=table_wf)

• All log models

# build all log model
model_wf_all_log <- lm(log(y) ~ log(X1) + log(X2) + log(X3) + log(X4) + log(X5) + log(X6) + log(X7) + log(X8) + log(X9), data=table_wf)
ols_vif_tol(model_wf_all_log)

model_wf_all_log_inter <- lm(log(y) ~ (log(X1) + log(X2) + log(X3) + log(X4) + log(X5) + log(X6) + log(X7) + log(X8) + log(X9))^2, data=table_wf)
model_wf_rm2457_all_log <- lm(log(y) ~ log(X1) + log(X3) + log(X6) + log(X8) + log(X9), data=table_wf)
model_wf_rm257_all_log <- lm(log(y) ~ log(X1) + log(X3) + log(X4) + log(X6) + log(X8) + log(X9), data=table_wf)
model_wf_rm247_all_log <- lm(log(y) ~ log(X1) + log(X3) + log(X5) + log(X6) + log(X8) + log(X9), data=table_wf)
model_wf_rm125_all_log <- lm(log(y) ~ log(X3) + log(X4) + log(X6) + log(X7) + log(X8) + log(X9), data=table_wf)

ols_vif_tol(model_wf_aic_all_log)

# Comparison
huxreg(model_wf_aic_log, model_wf_aic_all_log,model_wf_rm2457_all_log, model_wf_rm247_all_log, model_wf_rm257_all_log, model_wf_rm125_all_log, model_wf_aic_log_inter, model_wf_aic_all_log_inter)

• Mixed models

# Mixed regression 1
model_wf_mix1 <- lm(log(y) ~ (X1 + X3 + X4 )^2 + log(X8) + log(X9), data=table_wf)

# Mixed regression 2
model_wf_mix2 <- lm(log(y) ~ (log(X1) + log(X3) + log(X4) )^2 + log(X8) + log(X9), data=table_wf)

# Mixed regression 3
model_wf_mix3 <- lm(log(y) ~ (X1 + X3 + X4 )^2 + log(X2)+log(X5)+log(X6)+log(X7) + log(X8) + log(X9), data=table_wf)

# Comparison
huxreg(model_wf_aic_mix1, model_wf_aic_mix2, model_wf_aic_mix3)

Additional Steps

• step both

# Stepwise Regression based on p values for full model#
k <- ols_step_both_p(model_wf_full_log)

## Stepwise Selection Method   
## ---------------------------
## 
## Candidate Terms: 
## 
## 1. X1 
## 2. X2 
## 3. X3 
## 4. X4 
## 5. X5 
## 6. X6 
## 7. X7 
## 8. X8 
## 9. X9 
## 
## We are selecting variables based on p value...
## 
## Variables Entered/Removed: 
## 
## - X4 added 
## - X3 added 
## - X7 added 
## 
## No more variables to be added/removed.
## 
## 
## Final Model Output 
## ------------------
## 
##                         Model Summary                         
## -------------------------------------------------------------
## R                       0.944       RMSE               0.549 
## R-Squared               0.890       Coef. Var          8.618 
## Adj. R-Squared          0.878       MSE                0.301 
## Pred R-Squared          0.854       MAE                0.414 
## -------------------------------------------------------------
##  RMSE: Root Mean Square Error 
##  MSE: Mean Square Error 
##  MAE: Mean Absolute Error 
## 
##                                ANOVA                                
## -------------------------------------------------------------------
##                Sum of                                              
##               Squares        DF    Mean Square      F         Sig. 
## -------------------------------------------------------------------
## Regression     63.565         3         21.188    70.378    0.0000 
## Residual        7.828        26          0.301                     
## Total          71.393        29                                    
## -------------------------------------------------------------------
## 
##                                  Parameter Estimates                                  
## -------------------------------------------------------------------------------------
##       model     Beta    Std. Error    Std. Beta      t       Sig      lower    upper 
## -------------------------------------------------------------------------------------
## (Intercept)    2.872         0.547                 5.254    0.000     1.748    3.995 
##          X4    0.122         0.033        0.559    3.730    0.001     0.055    0.189 
##          X3    0.168         0.040        0.435    4.165    0.000     0.085    0.251 
##          X7    3.106         1.537        0.309    2.021    0.054    -0.053    6.266 
## -------------------------------------------------------------------------------------

# k
# plot(k)

# Stepwise AIC Regression for full model#
k<- ols_step_both_aic(model_wf_full_log)

## Stepwise Selection Method 
## -------------------------
## 
## Candidate Terms: 
## 
## 1 . X1 
## 2 . X2 
## 3 . X3 
## 4 . X4 
## 5 . X5 
## 6 . X6 
## 7 . X7 
## 8 . X8 
## 9 . X9 
## 
## 
## Variables Entered/Removed: 
## 
## - X4 added 
## - X3 added 
## - X7 added 
## - X8 added 
## - X9 added 
## - X6 added 
## 
## No more variables to be added or removed.

# k
# plot(k)

# Stepwise Regression based on p values for all log model #
k <- ols_step_both_p(model_wf_all_log)

## Stepwise Selection Method   
## ---------------------------
## 
## Candidate Terms: 
## 
## 1. log(X1) 
## 2. log(X2) 
## 3. log(X3) 
## 4. log(X4) 
## 5. log(X5) 
## 6. log(X6) 
## 7. log(X7) 
## 8. log(X8) 
## 9. log(X9) 
## 
## We are selecting variables based on p value...
## 
## Variables Entered/Removed: 
## 
## - log(X4) added 
## 
## No more variables to be added/removed.
## 
## 
## Final Model Output 
## ------------------
## 
##                         Model Summary                         
## -------------------------------------------------------------
## R                       0.954       RMSE               0.479 
## R-Squared               0.910       Coef. Var          7.526 
## Adj. R-Squared          0.907       MSE                0.230 
## Pred R-Squared          0.896       MAE                0.353 
## -------------------------------------------------------------
##  RMSE: Root Mean Square Error 
##  MSE: Mean Square Error 
##  MAE: Mean Absolute Error 
## 
##                                ANOVA                                 
## --------------------------------------------------------------------
##                Sum of                                               
##               Squares        DF    Mean Square       F         Sig. 
## --------------------------------------------------------------------
## Regression     64.964         1         64.964    282.937    0.0000 
## Residual        6.429        28          0.230                      
## Total          71.393        29                                     
## --------------------------------------------------------------------
## 
##                                  Parameter Estimates                                  
## -------------------------------------------------------------------------------------
##       model     Beta    Std. Error    Std. Beta      t        Sig     lower    upper 
## -------------------------------------------------------------------------------------
## (Intercept)    4.189         0.156                 26.803    0.000    3.868    4.509 
##     log(X4)    1.259         0.075        0.954    16.821    0.000    1.106    1.413 
## -------------------------------------------------------------------------------------

# k
#plot(k)

# Stepwise AIC Regression for all log model #
k <- ols_step_both_aic(model_wf_all_log)

## Stepwise Selection Method 
## -------------------------
## 
## Candidate Terms: 
## 
## 1 . log(X1) 
## 2 . log(X2) 
## 3 . log(X3) 
## 4 . log(X4) 
## 5 . log(X5) 
## 6 . log(X6) 
## 7 . log(X7) 
## 8 . log(X8) 
## 9 . log(X9) 
## 
## 
## Variables Entered/Removed: 
## 
## - log(X4) added 
## - log(X8) added 
## - log(X9) added 
## - log(X6) added 
## - log(X1) added 
## - log(X3) added 
## 
## No more variables to be added or removed.

# k
# plot(k)

# Stepwise Regression based on p values for full log model #
# k <- ols_step_both_p(model_wf_full_log_inter)
# k
# plot(k)

# Stepwise AIC Regression for all full model #
# k <- ols_step_both_aic(model_wf_full_log_inter)
# k
# plot(k)

# Stepwise Regression based on p values for all log model #
# k <- ols_step_both_p(model_wf_all_log_inter)
# k
# plot(k)

# Stepwise AIC Regression for all log model #
# k <- ols_step_both_aic(model_wf_all_log_inter)
# k
# plot(k)

# Stepwise Regression based on p values for all log model #
# k <- ols_step_both_p(model_wf_mix2 )
# k
# plot(k)

# k <- ols_step_both_aic(model_wf_mix2)
# k
# plot(k)

# Stepwise Regression based on p values for X4 eliminated model#
# k <- ols_step_both_p(model_wf_rm4_log)
# k
# plot(k)

# Stepwise AIC Regression for X4 eliminated model#
# k<- ols_step_both_aic(model_wf_rm4_log)
# k
# plot(k)

# Stepwise Regression based on p values for X1 eliminated model#
# k <- ols_step_both_p(model_wf_rm1_log)
# k
# plot(k)

# Stepwise AIC Regression for X1 eliminated model#
# k<- ols_step_both_aic(model_wf_rm1_log)
# k
# plot(k)

# All Possible Regression for full log model #
# k <- ols_step_all_possible(model_wf_full_log)
# plot(k)
# head(arrange(k, desc(adjr)))

# All Possible Regression for all log model #
# k <- ols_step_all_possible(model_wf_all_log)
# plot(k)
# head(arrange(k, desc(adjr)))

# All Possible Regression for 3g log model #
#!!!!!!!!!!!! k <- ols_step_all_possible(model_wf_3g_log_inter)
# plot(k)
# head(arrange(k, desc(adjr)))
 
# All Possible Regression for mixed log model #
# k <- ols_step_all_possible(model_wf_mix2 )
# plot(k)
#  head(arrange(k, desc(adjr)))

# All Possible Regression for X4 eliminated model #
# k <- ols_step_all_possible(model_wf_rm4_log)
# k
# plot(k)

# All Possible Regression for X1 eliminated model #
# k <- ols_step_all_possible(model_wf_rm1_log)
# k
# plot(k)

#Lack of Fit F Test

ols_pure_error_anova(lm(y~X1, data = table_wf))
ols_pure_error_anova(lm(y~X4, data = table_wf))

alias(lm(y ~ as.factor(X3) + as.factor(X4) + as.factor(X5) + as.factor(X6) + as.factor(X7), data=table_wf))

alias(lm(y ~ as.factor(X1) + as.factor(X8) , data=table_wf))

alias(lm(y ~ as.factor(X4) + as.factor(X9) , data=table_wf))

alias(lm(y ~ as.factor(X3) + as.factor(X6) + as.factor(X7) + as.factor(X8) + as.factor(X9) , data=table_wf))

Final models

• Check model all log 13689

summary(model_wf_rm2457_all_log)

## 
## Call:
## lm(formula = log(y) ~ log(X1) + log(X3) + log(X6) + log(X8) + 
##     log(X9), data = table_wf)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -0.71392 -0.10557  0.06481  0.11154  0.25306 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  5.87592    0.22021  26.684  < 2e-16 ***
## log(X1)      0.68118    0.02343  29.067  < 2e-16 ***
## log(X3)      0.37527    0.09489   3.955 0.000591 ***
## log(X6)     -0.33576    0.08150  -4.120 0.000389 ***
## log(X8)      1.72499    0.16768  10.288 2.82e-10 ***
## log(X9)     -1.45653    0.14975  -9.727 8.42e-10 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.2148 on 24 degrees of freedom
## Multiple R-squared:  0.9845, Adjusted R-squared:  0.9813 
## F-statistic: 304.6 on 5 and 24 DF,  p-value: < 2.2e-16

Anova(model_wf_rm2457_all_log)

ols_regress(model_wf_rm2457_all_log)

##                         Model Summary                         
## -------------------------------------------------------------
## R                       0.992       RMSE               0.215 
## R-Squared               0.984       Coef. Var          3.374 
## Adj. R-Squared          0.981       MSE                0.046 
## Pred R-Squared          0.974       MAE                0.141 
## -------------------------------------------------------------
##  RMSE: Root Mean Square Error 
##  MSE: Mean Square Error 
##  MAE: Mean Absolute Error 
## 
##                                ANOVA                                 
## --------------------------------------------------------------------
##                Sum of                                               
##               Squares        DF    Mean Square       F         Sig. 
## --------------------------------------------------------------------
## Regression     70.285         5         14.057    304.557    0.0000 
## Residual        1.108        24          0.046                      
## Total          71.393        29                                     
## --------------------------------------------------------------------
## 
##                                   Parameter Estimates                                    
## ----------------------------------------------------------------------------------------
##       model      Beta    Std. Error    Std. Beta      t        Sig      lower     upper 
## ----------------------------------------------------------------------------------------
## (Intercept)     5.876         0.220                 26.684    0.000     5.421     6.330 
##     log(X1)     0.681         0.023        0.908    29.067    0.000     0.633     0.730 
##     log(X3)     0.375         0.095        0.124     3.955    0.001     0.179     0.571 
##     log(X6)    -0.336         0.081       -0.109    -4.120    0.000    -0.504    -0.168 
##     log(X8)     1.725         0.168        0.543    10.288    0.000     1.379     2.071 
##     log(X9)    -1.457         0.150       -0.533    -9.727    0.000    -1.766    -1.147 
## ----------------------------------------------------------------------------------------

confint(model_wf_rm2457_all_log, level = 1-(0.05/6))

##                0.417 %   99.583 %
## (Intercept)  5.2428046  6.5090281
## log(X1)      0.6137985  0.7485522
## log(X3)      0.1024526  0.6480888
## log(X6)     -0.5700799 -0.1014438
## log(X8)      1.2429032  2.2070673
## log(X9)     -1.8870607 -1.0259933

# Collinearity Diagnostics #
 ols_vif_tol(model_wf_rm2457_all_log)

#Model Fit Assessment
 ols_plot_diagnostics(model_wf_rm2457_all_log)

# Part & Partial Correlations
 ols_test_correlation(model_wf_rm2457_all_log) # Correlation between observed residuals and expected residuals under normality.

## [1] 0.9123016

# Residual Normality Test
 ols_test_normality(model_wf_rm2457_all_log) # Test for detecting violation of normality assumption. #If p-value is bigger, then no problem of non-normality #

## -----------------------------------------------
##        Test             Statistic       pvalue  
## -----------------------------------------------
## Shapiro-Wilk              0.846           5e-04 
## Kolmogorov-Smirnov        0.163          0.3630 
## Cramer-von Mises          6.9863         0.0000 
## Anderson-Darling          1.1772         0.0038 
## -----------------------------------------------

# Variable Contributions
 ols_plot_added_variable(model_wf_rm2457_all_log)

# Residual Plus Component Plot
 ols_plot_comp_plus_resid(model_wf_rm2457_all_log)

• Check model mix1

summary(model_wf_mix1)

## 
## Call:
## lm(formula = log(y) ~ (X1 + X3 + X4)^2 + log(X8) + log(X9), data = table_wf)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -0.58067 -0.07614 -0.01057  0.08042  0.34122 
## 
## Coefficients: (1 not defined because of singularities)
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  2.59769    0.22800  11.394 1.07e-10 ***
## X1           1.28956    0.23173   5.565 1.36e-05 ***
## X3           0.29604    0.03971   7.456 1.86e-07 ***
## X4           0.39072    0.04174   9.361 3.96e-09 ***
## log(X8)      1.57550    0.17201   9.159 5.82e-09 ***
## log(X9)     -1.34486    0.15424  -8.719 1.37e-08 ***
## X1:X3       -0.38088    0.05579  -6.827 7.38e-07 ***
## X1:X4        0.03070    0.01243   2.471   0.0217 *  
## X3:X4             NA         NA      NA       NA    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.2266 on 22 degrees of freedom
## Multiple R-squared:  0.9842, Adjusted R-squared:  0.9791 
## F-statistic: 195.5 on 7 and 22 DF,  p-value: < 2.2e-16

Anova(model_wf_mix1)

# Collinearity Diagnostics #
ols_vif_tol(model_wf_mix1)

#Model Fit Assessment
ols_plot_diagnostics(model_wf_mix1)

# Part & Partial Correlations
ols_test_correlation(model_wf_mix1) # Correlation between observed residuals and expected residuals under normality.

## [1] 0.9683687

# Residual Normality Test
ols_test_normality(model_wf_mix1) # Test for detecting violation of normality assumption. #If p-value is bigger, then no problem of non-normality #

## -----------------------------------------------
##        Test             Statistic       pvalue  
## -----------------------------------------------
## Shapiro-Wilk              0.9448         0.1225 
## Kolmogorov-Smirnov        0.1306         0.6387 
## Cramer-von Mises          6.871          0.0000 
## Anderson-Darling          0.5822         0.1182 
## -----------------------------------------------

# Variable Contributions
ols_plot_added_variable(model_wf_mix1)

# Residual Plus Component Plot
ols_plot_comp_plus_resid(model_wf_mix1)

• Check model all log 134689

 summary(model_wf_rm257_all_log)

## 
## Call:
## lm(formula = log(y) ~ log(X1) + log(X3) + log(X4) + log(X6) + 
##     log(X8) + log(X9), data = table_wf)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -0.70374 -0.09978  0.03781  0.11788  0.24638 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  5.29728    0.49196  10.768 1.86e-10 ***
## log(X1)      0.46313    0.16797   2.757 0.011217 *  
## log(X3)      0.24862    0.13447   1.849 0.077357 .  
## log(X4)      0.41702    0.31819   1.311 0.202940    
## log(X6)     -0.32609    0.08065  -4.043 0.000505 ***
## log(X8)      1.71460    0.16541  10.366 3.87e-10 ***
## log(X9)     -1.45039    0.14763  -9.824 1.07e-09 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.2117 on 23 degrees of freedom
## Multiple R-squared:  0.9856, Adjusted R-squared:  0.9818 
## F-statistic: 261.7 on 6 and 23 DF,  p-value: < 2.2e-16

 Anova(model_wf_rm257_all_log)

# Collinearity Diagnostics #
# ols_vif_tol(model_wf_rm257_all_log)

#Model Fit Assessment
# ols_plot_diagnostics(model_wf_rm257_all_log)

# Part & Partial Correlations
# ols_test_correlation(model_wf_rm257_all_log) # Correlation between observed residuals and expected residuals under normality.

# Residual Normality Test
# ols_test_normality(model_wf_rm257_all_log) # Test for detecting violation of normality assumption. #If p-value is bigger, then no problem of non-normality #

# Variable Contributions
# ols_plot_added_variable(model_wf_rm257_all_log)

# Residual Plus Component Plot
# ols_plot_comp_plus_resid(model_wf_rm257_all_log)

• Check model all log 135689

summary(model_wf_rm247_all_log)

## 
## Call:
## lm(formula = log(y) ~ log(X1) + log(X3) + log(X5) + log(X6) + 
##     log(X8) + log(X9), data = table_wf)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -0.67722 -0.08003  0.01102  0.13879  0.25715 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  0.57120    3.36000   0.170  0.86650    
## log(X1)      0.72550    0.03608  20.107 4.31e-16 ***
## log(X3)      0.41866    0.09605   4.359  0.00023 ***
## log(X5)      1.25873    0.79566   1.582  0.12731    
## log(X6)     -0.26702    0.09022  -2.960  0.00702 ** 
## log(X8)      1.62253    0.17508   9.267 3.15e-09 ***
## log(X9)     -1.37489    0.15416  -8.919 6.33e-09 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.2084 on 23 degrees of freedom
## Multiple R-squared:  0.986,  Adjusted R-squared:  0.9824 
## F-statistic: 270.1 on 6 and 23 DF,  p-value: < 2.2e-16

Anova(model_wf_rm247_all_log)

# Collinearity Diagnostics #
# ols_vif_tol(model_wf_aic_all_log_inter)

#Model Fit Assessment
# ols_plot_diagnostics(model_wf_aic_all_log_inter)

# Part & Partial Correlations
# ols_test_correlation(model_wf_aic_all_log_inter) # Correlation between observed residuals and expected residuals under normality.

# Residual Normality Test
# ols_test_normality(model_wf_aic_all_log_inter) # Test for detecting violation of normality assumption. #If p-value is bigger, then no problem of non-normality #

# Variable Contributions
# ols_plot_added_variable(model_wf_aic_all_log_inter)

# Residual Plus Component Plot
# ols_plot_comp_plus_resid(model_wf_aic_all_log_inter)